A badge in the shape of the Children of the Cathedral symbol. On one side you notice bumps and indentations, almost reminding you of a key.”
COC badge (black) is a miscellaneous item in Fallout.
Behind the locked door in the Cathedral that requires a different badge is a room. On the east side of that room is another locked door. Like the door before, this one can be unlocked by using lockpicking or by using this badge.